On the rate of convergence of certain summability methods for Fourier integrals of \(L^ 2\)-functions
Let \(B\) be a bounded convex symmetric body in \(R^ n\), and \(|\cdot|\) the Minkowski norm associated to \(B\), \(| x|=\inf\{t>0\mid t^{-1}x\in B\}\), \(x\in R^ n\). The authors consider the Fourier multiplier associated to a bounded function \(m\in L^ \infty(0,\infty)\) by \((\widehat{T_{m_ t}}f)(\xi)=m_ t(|\xi|)\hat f(\xi)\), where \(m_ t(s)=m(ts)\). The norms on the function spaces considered are defined by fractional derivatives which are given in terms of \(I^ \alpha_ \omega\), where \[ I^ \alpha_ \omega m(t)=(1/\Gamma(\alpha))\int^ \omega_ t(s-t)^{\alpha- 1}m(s)ds,\;0<t<\omega;\quad I^ \alpha_ \omega m(t)=0,\quad t\geq\omega. \] The fractional derivative is defined by \(m^{(\alpha)}(t)=\lim_{\omega\to\infty}\left(-{d\over dt}I_ \omega^{1-\alpha}m(t)\right)\). The first function space considered is \[ RL(2,\alpha)=\{m\in L^ \infty(0,\infty):\| m\|_{RL(2,\alpha)}=\sup_{t>0}\|(\chi m_ t)^{(\alpha)}\|_ 2<\infty\}, \] where \(\chi\) is any nontrivial bump function. The results can also be expressed in terms of \(WBV_{2,\alpha}\) which is the set of \(m\in L^ \infty(0,\infty)\cap C(0,\infty)\) such that \(m^{(\alpha)}\) exists and \(\| m\|_{2,\alpha}=\| m\|_ \infty+\sup_{k\in Z}\left\{\int^{2^ k}_{2^{k-1}}| t^ \alpha m^{(\alpha)}(t)|^ 2dt/t\right\}^{1/2}\) is finite. \textit{A. Carbery}, \textit{G. Gasper} and \textit{W. Trebels} [J. Approximation Theory 48, 251-261 (1986; Zbl 0602.42020)] have proved that \(RL(2,\alpha)=WBV_{2,\alpha}\) for \(\alpha>1/2\). The authors consider a doubling function \(\varphi\) on \([0,\infty]\) which is a nondecreasing function such that \(1\leq\varphi(2t)\leq\lambda\varphi(t)\), where \(\lambda\geq 1\). Denote \(\mu=\log_ 2(\lambda)\). They let \(L^ 2_ \varphi=\bigl\{f\mid\bigl(\int_{R^ n}|\hat f(\xi)|^ 2|\varphi(|\xi|)|^ 2d\xi\bigr)^{1/2}<+\infty\bigr\}\) with the obvious norm. Their main result is: If \(\alpha>1/2\), \(m\in RL(2,\alpha)\), and for some \(\beta>\mu\) and \(\gamma>1\), \[ \|(\chi m_ t)^{(\alpha)}\|_ 2=O(t^ \beta)\quad\text{as } t\to 0+, \tag{1} \] \[ \|(\chi m_ t)^{(\alpha)}\|_ 2=O((\log t)^{- \gamma})\quad\text{as } t\to\infty, \tag{2} \] then \[ \|\sup_{t>0}| T_{m_ t}f|\varphi(1/t)\|_ 2\leq C\| f\|_{L^ 2_ \varphi}. \] They give more precise versions saying that roughly condition (1) controls the behavior at 0 of the left- hand side while condition (2) controls its behavior at infinity. Using the estimates, they give results on the pointwise convergence of \(T_{m_ t}f\) to \(f\) with rates of convergence depending on \(\varphi\) for \(f\in L^ 2_ \varphi\). I will state one such result. If \(m\in RL(2,\alpha)\), \(\alpha>1/2\), and condition (1) holds for \(m-1\) with \(\beta>\mu\), then, for every \(f\in L^ 2_ \varphi\), \(T_{m_ t}f- f=o(1/\varphi(1/t))\) a.e. as \(t\to 0+\).
scientific article; zbMATH DE number 60552